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11. CONCOOLUNG ORGAIZAIO NAME AND ADORES$ 0 SOME1IMT wt.TS

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I&. SUIPPLEMEINTARY MOTES

It. KEY WORDS JAanf'sa. W, I.WW VW #f n-=y POW ~Igrnf*Y by b~k* a*W.6a~w)

Hi.gh-Energy Lasers HZL PropagationCon~tinuous-Wave Thk~rznal Blooming HEL System Analysis

XL ~ ~~S ANTAtM- ",V &w' - ,tpy by hi, amawJ

This pa~per prescnts an algebraic m~odel for c-nti;ý.-ous-wave thegmal-blooming eifects that is accurate enough to well represent a large wave-optics data base, simple "nough to suggest same previously unnoticed

CL universal relationships and complete enough to allow ivariation of themany physical variablesfof interest in asytema analysis exercises.=

DOanI 1473 41~onw ou I Nov uas a 08 TI / UNCLaSSIFIED3 IMMITYCLAISPOCATION1 OP THIS PAGEMN DWif heXUIM0i- -.4 r

Vk~ft

R-1173

AN ALGEBRAIC MODEL

FOR CONTINUOUS-WAVETHERMAL-BLOOMING EFFECTS

by

H. BreauxBallistic Research Laboratozy

Aberdeen Proving Ground, Maryland

W. Evers and R. SepuchaU.S. Army High Energy Laser Systems Project Office

Redstone Arsenal, Alabama

C. WhitneyThe Charles Stark Draper Laboratory, Inc.

Cambridge, Massachusetts

July 1978

Approved: , 7T V7&'

I

The Charles Stark Draper Laboratory, Inc.Cambridge, Massachusetts 02139

ACKNOwlEDGMENT

This report was prepared by The Charles Stark Draper Laboratory,

Inc. under Contracts DAAK40-76-C-1145 and DAAK40-78-C-0117 with the U.S.

Army High Energy Laser Systems Project Office, Huntsville, Alabama.

Publication of this report does not constitute approval by the

U.S. Army of the findings or conclusions contained heroin. It is

published for the exchange and stimulation of ideas.

3AA

*1

I

II

[.1

iiI

Kr

TABLE OF CONTENTS

section Pg

1 INTRODUCTION .................................................... 1

2 INTENSITY DEGRADATION AS A FUNCTION OF POWER.................. 4

3 PHYSICAL. MODEL.................................................. 8

4 PHASE INTEGRAL................... ............................. 13

5 CORRELATION DATA............................................... 17

6 SUMMARY OF FORMULAS FOR SYSTEM4S ANALYSIS..................... 23

LIST OF REFERENCES...................................................... 26I

SECTION 1

INTRODUCTION

The purpoce of this document is to provide a standard algebraic

model for a continuous-wave (CW) high-energy-laser (HEL) beam on target.

The betnt is degraded by thermal blooming as well as numerous other

effects. The algebraic model is intended for systems analysis exer-

cises, where the numerous parameters are to be explored, making more

detailed computer simulation of atmospheric propagation impractical.

It is an outgrowth of efforts to fit scaling laws to a large body of

data generated by detailed atmospheric propagation simulations using

a finite-difference wave-optics code.*

Any such wave-optics code provides a detailed intensity profile

on target, from which several different summary characteristics can be

extracted. These include, for instance, peak intensity, line-of-sightbeam dispersion, and beam area. The last of these further admits

several definitions, including area to some percentage of total power,

and area defined by a ratio of squared integral of intensity to integralof intensity squared (suggested independently by Lincoln Laboratoryand Draper Laboratory researchers). If beams on target were Gaussianin shape, all such characteristics would convey equivalent information.The simplified algebraic model assumes that this is nearly the case,

and speaks nominally of peak intensity on target.

Regression analysiu of the results has shown that peak intensity

can be correlated with an integral T h' which represents the accumulation

along range (starting from the center of the tperture) of phase pertur-

bation due to heating in a beam with absorption, scattering, convective

clearing, and focusing. Figures 1, 2, and 3 show the tightness of the

correlation obtained for three different beam shapes. The ordinate is

the phase integral Th and the abscissa in the ratio R - (IU - [)/I,

where I is peak intensity and IU is unbloomed peak intensity.

Data provided by D. Cordray of Naval Research Lab (IRL).

l k

Any one of many numerically similar functional forms could be fitto the curves defined by the correlation data. Thir document selectsone that has been found most uceful bocause of its algebraic simplicity.Section 2 shown how the simple algebraic form leads to a number ofuniversal relationshipii that are independent of almost all physicaldetails of the propagation process. These relationshiis describe thevariation of peak intensity as a function of pover alone, with allother variables held constant. Important relationships are shown todepend only on beam shape. This conclusion has not previously beenevident from other scaling laws.

Section 3 discusses the parameters appearing in Section 2, showinghow each depends on actual physical variables that describe the mgage-ment, the laser, and the atmosphere. Section 4 provides a techniquefor accurately evaluating the phase integral along range required inSection 3. For several coinon beam shapes, Section 5 discusses numericalvalues of regression parameters defined in Section 3. Section 6 pro-vides a concise summary of all formulas required for systems analysis

exercises.

.4.5

3.5K j1.5

3K9

10 X

*h

Figure 1. Correlation for infiniteGaussian beam.

2

FF4.0 u " I

, it 'I

4.0

.0

2.0 -

0 10 20 30 40 50S9h

Figure 2. Correlation for i/e truncatedGaussian~ beam.

5 11 I J I I I I

5K

4

33K

1KK

10 30 50 70 90 110 130 10•h

Figure 3. Correlation for uniform beam,

.3

_______

SELTION 2

INTENSITY DEGRADATION AS AFUNCTION OF POWER

For a class of problems differing only in time-average power (P),the ratio of bloomed to unbloomed intensity focused on target can be

modeled au

20L a

2Bin this expression, a is the 1/e line-of-sight beam dispersion in

radians due to linear effects, including diffraction, beam quality,turbulence, and jitter. The l .atter two effects separato into 'high"and "low" frequencies, whicn, respectively, do and do not impactblooming. The latter is represented in the term 2B. To some exant,

the form of I(P)/Iu(P) expresses the familiar idea of Orss-ing" (root-sum-squaring). For a Gaussian beam, dispersive effects combine bysumming variances, and the denominator (a + a2) resembles such a sum.L B

2The blooming term (a ) dependa on power (P) in a way that can be

modeled by a variety of functional forms. The choice of form is a

tradeoff between simplicity and range of validity. In this report,we use the simplest form known to be valid for power levels of practicalinterest. This form is nominally

2 aB• - CBPa

aThe use of P with a > 1 allows reproduction of a well-knownphysical pheaomenons there exists a critical power PC sach that the

This type of foriaula has also been suggested and used by F. Gothardtand J. Wallace. In particular, Wallace suggested a - 3/2.

-

"• ! .4

2intensity on target is maximum. That PC can be related to aL, CB,

and a am follows. The unbloomed intensity on target is proportional

to P, so that actual intensity is proportional to orPC:/(aT + CBP ). Differentiating with respect to P and setting thederivative to zero gives

[2]l/aCa

S]1

BI'From the value of PC, the ratio I(Pc)/Iu(Pc) is readily found

to be

I (P C) a-

This result is interesting because it depends only on the parameter a,2expressive rf beam shape alone, and not on a or C3, which contain much

of the physics of the problem. It therefore presents a physicalphenomenon that is essentially separable from other physical phenomenain the overall propagation process.

The result concerning I(Pc)/Iu(Pc) indicates what to expect from

adaptive phase correction for thermal blooming. Since phase corroction idoes nt change beam shape, it will move the whole curve of intensity-out versus power-in in such a way that the new peak, the old peak, and

the origin lie on a straight line, as illustrated in Figure 4.

Clearly, PC marks the upper limit of power levels having practicalinterest. In fact, operation well below PC may be of interest, so letus consider P at same fraction of PC: P - Pc/b° Then

2 aI a L + C a(LC)C b a a + c IC

which simplifies to

WC7 b&(a 1) + 1

This result, too, is independent of a and C.of and therefore is

independent of the phenomena controlling them.

5

SLOPE 1

SLOPf E

Figure 4. Adaptive correction of phase withoutchange of beam shape.

The formula describing opezation below PC is plotted in Figure 5for several typical values of a. For all cases, there is little pointin operating at power levels above PC/2, because more than 85% of thelimiting peak intensity is already available at PC/2.

The formulas for I(Pc)/Iu(Pc) and I(Pc/b)/I(Pc) can be combinedto relate I(PM b) to I (P ). The result is

I()(a -1)b

- aU ba(a - 1) + 1

2This result, too, is independent of the parameters aL and C3 that carry

most of the physics of the prohlem. It means essentially that if Iu(PC)is specified, I for any other condition is implicitly specified.

S6j

• L • . . .. .

S1.0

0

FRAMTON OF PC

Figure 5. Operation below PC

C..1 1

S~The model presented in this section in most accurate in the regime

of physical interest for systems modeling, namely, below the zero-wander

value of PC. This in the came because below this power there in relatively

little blooming, with little attendant beam distortion and no beam-•

breakup. There in then a one-to-one Gaussian-like correspondence between

intensity I and the various disr,-rsiva a 2 terms. Operation far above

such a power level requires a mor,.* complex model,, involving linear and

quadratic laser power terms in aB presented by Breaux. (W*

The ratio-type results of this section are of a very general

nature, and are valid regardless of the methodology used to evaluat.e a,2

oL CBI or lu(Pc). Specific techniques for evaluating these are

S~provided in the Sections 3 through S.

S~Superscript numerals refer to similarly numbered items in the Li~sti of References.

a - 1.r

'Ai

SECTION 3

PHYSICAL MODEL

The model for degradation of intensity focused on target as a2function o. power has only three parameters (a, 0 L, and C.), and requires

only IU(P)C to specify intensity under any conditions. A small amount

of data for a given class of problems differing only in P will suffice

to determine all the required quantities. But extrapolating the rp-

sults to any other class of problems requires some kind of physical

model. The expoi.ent a can be assumed to be independent of many physical

variables, so the more pressing problems are to admit defocus, and to

model a', C,, and Iu(P

The physical variables affecting HEL propagation are designated

in Table 1. In cases where t.he propagation path extends through signif-

icantly different altitudes, path averaging of parameters is required.2(See Reference 1, p. 59 for applicable techniques regarding C2.)n

These parameters can be summarized in terms of four dimensionless

numbers:

Absorption number NA -OLZ

• Ut t

Slew number WS "U

21iR 2

Fresnel number NF " 2i tra

Distortion number ND - _W|-3

Defined appropriate to this model.

Subscript - me,'ns unperturbed natural value.

18

I.ii_*H

Table 1. Physical variables affectingHEL propagation.

Parameter Variable Units

Engagemert

Target range Zt meters

Transverse target Ut meters/secondvelocity

Laser

Aperture radius Rm meters

Time average power P Joules/second

Wavelength meters

Beam quality M no dimension

Beam shape no dimension, not a scalar

Atmosphere

Wind velocity Uw meters/second

Turbulence C2 meters-2/3n

Absorption a meters

Scattering a meters-1

Index of refraction n no dimension

Temperature T degrees

Refraction gradient 3n/3T degrees 1

Density p kilogram-meters-3

Heat capacity Cp Joules/degree-kilogram

IP

9_ _ _ A

There are also effective beam qualities N0 and N6, which repre-

sent the actual se-oading of the beam due to various effects. Theseefiects include beam shape and the blooming itself, and so NQ and

N6 must be defined later. A procedure is provided in Section 6.

The ro2 is l/e radial beam dispersion due to linear effects. ItL

includes a number of contributors, distinguished by different subscripts:diffraction and beam quality (D), high--frequency beam distortion and

motion due to turbulence (T) and jitter (J), low-frequency beam wanderMW) due to turbulence, jitter, and pointer-tracker effects. Thus, we

have

2 2 +a 2+ 2+ 2L D T "3 W

The formula for the diffraction contributor is

a - O.5(I.It--2

where m' is characteristic of beam shape. Requiring T aD to be the

63% beam radius on target makes m' equal to 2/w for an infinite

Gaussian beam, equal to 0.9166 for a 1/e2 truncated Gaussian beam, and

equal to 0.9202 for a uniform beam (see Reference 2).

The turbulence contributor refers specifically to the short-termhigh-freqW~ency part of total turbulence. It has been found by Breaux(3)

that for a large variety of beam shapes, total turbulence is well

represented by

a2)2 2(D)2

aTT VW rM 0r

where r 0 is the Fried coherence length for wave number k - 2w/A

r t -3/5

r 2.10 45k2 Z (f) d2M00

10

F and De is an effective aperture size, appropriate to the beam shape. For

infinite Gaussian, truncated Gaussian, or uniform beams, reepectively,

it is

2 2 2 2 IDe 8R; Sm 3 .7%, 4R;S~e

The short-term part (aT is smaller, varying between the values

T

for De/r0 < 3 and 0.182iR02 CA))-[(L)- \r0,)J

S~To ensure model validity, the high-frequency turbulence should be

limited to values small enough to cause no speckling, say uT< 2oD. The2 2 2fTT aT) may or may not appear in the wander term

depending on the particular hardware implementation being modeled.

Additional jitter and wander contributors arise from the particular

hardware considered, and must be set by the user.

The dispersion due to blooming is to be combined with the linear2 2 2 2

dispersions simplified by a2, c2 oa , and we proceed now to

consider the blooming term

2 a

The physical phenomenon that causes ca is accumulation along range of phaseB

due to heating, which increases with power (P). and decreases, to some2extent, with that portion of a that actually experiences the blooming,

11

I!

2 2namely a - O.N It has been found possible to define a measure Th forheating phase such that the following expanded expression vell represents

2

2- C (o - 2 a

where C' is a dimensionless coefficient that depends on beam shape.

Clearly

, 2 2 iCB "Ci(•- )

The form of Th that makes the above representation of a. possibleis proportional to NDNF/.N. The phase integral (Th also has anadditional factor to make it into an integral along range that takesaccount of extinction, clearing, and focusing of the beam. Its evalua-tion is the subject of Section 4.

Next, Iu(P ) is easily estimated by considering total extinction(c - a + a), beam shape and quality, and the spreading due to linear

effects. The estimate is

PC exp('cZt)I U iP c ) 2 1vZ 2 0o 2B

Here, B is a beam-shape factor, which can be roughly interpreted as the

ratio of average intensity to "peak" intensity in the focal plane. Fora Gaussian beam, it is exactly 1/2, and for any realistic beam shape,it is quite close to 1/2.

I

I:pf SECTION 4

PHASE INTEGRAL

In Section 3, the problem of evaluating CB was reduced to the

problem of evaluating the phase integral h Many integration pro-(1, 4,5)cedures have been investigated, and most can be mad* numericallyadequate. The following is straightforward to explain. Let

S~1

T J Y (z) dz0

where

z Zzt I

N W exp(-CZ) dt I x 0 (z,t), y 0 (Z,t j

Yh(Z) FR.z)

S~Here, exp(-eZ) represents extinction. Te variable Re(z) represents

spot radius at :, approximated by ras-ing focus and diffraction effects.

RsLs) R- 1/ )/ 1

The term 2eR/Uw represents the clearing time at the aperture, end pro-

2vides a normalization for the time integration. P/wR provides anormalization for the intensity, which depends on time (t) through x 0and yo, which are both 0 for t - 0. For a beam slowed in the xdirection

Yo0 Z(t) - 0

13

but x0 (st) varies with the local clearing velocity. Assuming x 0 , Y0 Onondimensionalized by aperture radius PM

-tU (z)Xo(zt) - - =Z

with KJ(z) representing that velocity, approximated by

U.l) - Uwll + ( NS)vA

As an example, consider a Gaussian beam with amplitude

2 _2A 0 (x 0 ' YO exp(-x 0 -yO)

For the Gaussian profile, the normalized intensity

2I[x 0 (z't} yO(z't)] 2 It•. u I(Z)\

2P~R) R (2)

The time integral of normalized intensity is

[ (2) .1)Absorbing the A72 (1/2) in the overall C• relating to b6am shape, and

substituting for U.(z) and R (z), leaves the required z integral to

be just

SNDNF exp(-EZ) dz0N6(1 + zN.sl - Z)

iS

The integral does not appear to be amenable to direct analyticevaluation, so the options are numerical integration or analyticapproximation. Typically, numerical integration is difficult because

14

small steps are required to handle the rapid variation of the integrand.Therefore, analytic approximation is preforred. Probably the fewestapproximations are required if integration by parts is employed. In

that case we findi

Th [xp(-Ez)h + Czt exp(-eZ)y da

0

The indefinite integral

/2!!2 2 . 1/2

-+sZN) [cl - N)2 F

is available in an integral table by Klerer and Grossman. (6) Naturally,

the expression contains an integration constant. In principle, this

constant needs to be reset in such a way as to nullify the remainder

term• 1

SZt f exp(-cZ)Yl dz

0

This can be accomplished at least approximately by using TO(z) - T•(O/2)in place of TOz). This makes the expression for Th quite ccwplicat-A.

Utility is greatly served by reducing it to the case of small attenu

tion CZt, where the integration constant has no impact, and Th reducesto

N N1 - -tln (NS ( 1 +A) (Ns + 1)

h ~ exp~et -B -+N + I+C

where2 1/2

A - [(N + 1) + C

SB M+1+C

2NKCr1

The condition for this result to be valid can be stated as

i - +( - exp(-ez t)y.()

16

rrSECTION 5

CORRELATION DATA

Exten3ive simulations of atmospheric propagation have been per-formed at NRL using a finite-difference wave-optics code, and theseprovide the data base that estaolishes i strong correlation of theform suggested in this report relating intensity on target to theparameter Ih* There is very little scatter, and that which does existmay be attributed to one of two factors:

(1) Earlier correlation studies by Seiders(7) established thatthe spot size in the integrand denominator of Th shouldnot be Just the ideal diffraction-limited value, mainlybecause of the iterative effect of blooming upon itself.To account somewhat for this in a way that maintaimsreasonable simplicity, a free constant (m) scaling the

diffraction spot was introduced and evaluated by minimizingresiduals. Actually, such scaling must depend to sameexten." or physical variables, especially power. The idea.e iteration to admit variable m has been tried, but so

far has not been sufficiently successful to justify theeffort.

,2) The raw data generated were not all of the same form. Datafor the infinite Gaussian beam comprised peak intensities,wbireds those .o" the truncated Gaussian comprised 1/a2 area,and those for the uniform beam momprised the more complexarea measuro of the form

dx dy

Mhe latter functional (A) is currently thought to lead tothe tightest correlation. s4

17

2i

The resultra of analyzing three beam shapes-infinite Gaussian, i/e2

truncated Gaussian, and uniform beam---are presented in Tables 2 through

4 and plotted in Figures 6 through 8. Supposing that ND and N. are

fixed and a2 equals zero, let us compare the three beam types. WeN 2 adtarecall that CB is proportional to C'aL and that

PC __kW ICB

and

I(PC) a-lU C

The results tend to follow intuitively understandable patterns. First,

the parameter m affects the tightness of the correlation much less in

the case of the Gaussian beams than in the case of the uniform beam.

This is to be expected because blooming in the Gaussian case is driven

mainly by gradients near the aperture, whereas for the uniform bv;am such

gradients are minimized and blooming is driven by beam shape nearer

the focus, which is described by a. Secondly, the uniform bewa has, 2

the smalleat a and C /VO, resulting in by far the largest PC. The

large PC is to be expected because, with les gradient near the aperture,

there is less lensing effect even at high power. The mall a means,

however, that the PC region is very broad and I(Pc)/Iu(fc) is very Gmall.

The two Gaussian beams have anonalously different C;, a, and a, but

they turn out to have similar mall PC values. But bethautie a is larger

for these cases, I(PC)/Iu(PC) is somewhat larger.

The actual peak intensity or. target at PC depends on PC' on

I(PC)/IU(Pc) - (a-l)/a, and on the diffraction-limited spot radius r5

=PCC aL

S'C) a r

18 *

The spot radius for the three beam shapes, respectively, is proportionalto 2/w, 0.9166, and 0.9202. with the result that (P)for the truncatedGaussian is only half that of the infinite Gaussian. The value for the

but at the price of nearly six times the power (which may nevertheless be

acceptable).

19

Table 2. Infinite Gaussiant I(P)/IU(P) - (1 + C;*a)-1.

rum Maximum C.aBrror Error

0.142 0.949 0.011851 1.5029 1.0

0.109 0.667 0.010796 1.5891 1.50.102 0.614 0.010612 1.6189 1.75

Chosen °0.101 0.565 0.010590 1.6419 2.07fit

0.112 0.607 0.010919 1.6715 2.5

PC 20.903 I(PC)/Iu(PC) - 0.39V•5 I(PC) = 20.163

&5.

4.5

3.5 s

R 2.5

x

1.5

iti

0 10 20 30 40

Figure 6. Functional fit for infiniteGaussian beam.

20

Table 3. i/,2 truncated Gausuian: I(P)/Iu(P)- (1 + C;)"

rum Maximum q a mError Error

0.170 0.981 0.018600 1.4323 3.5S~~~~~Chosen. :4o .. 4-Cho1en 0.169 0.924 0.019023 1.450fit

0.172 0.874 0.019630 1.4646 4.5

0.178 0.828 0.020398 1.4743 5.0

0.187 0.788 0.021313 1.4805 5.5

0.198 0.752 0.022366 1.4938 6.0

P 27.780 I(Pc)/Iu(PC) - 0.31077 I(P 10.276

a 4.5

3.5

2.5 - K

1.5

K ,• 0.5

0 10 20 30 40 50

Figure 7. Functional fit for 1/e2 truncatedGaussian beam.

21

21 _ _ _ _ __ _ I

_____

Table 4. Uniform beam: I(P)/IU(P) - (1 + C;Th)

rms Maximum C a m

Error Error B

0.092 0.204 0.014732 1.1300 C.5

0.066 0.157 0.014395 0.1614 0.75Choseft [ 0 . 0 6 3 0.139 0.014264 1.1777 1.0

0.093 0.255 0.014335 1.1986 1.5

0.112 0.317 0.014474 1.2054 1.750.130 0.374 0.014660 1..2106 2.0

P 160.09 I(Pc)/IU(PC) 0.15089 I(P 28.526

5

3

10 30 5w 7' 0 10 130 16

*h

.•Figure S. Functional fit for uniform beam.

22

22 ij

I

SECTION 6

SUMMARY OF FORMULAS FOR SYSTEMS ANALYSIS

Suppose that one wishes to calculate peak intensity on target for

a given set of conditions. The required steps are:

(1) Choose beam shape and set C•, a, m, m', and m".

SC; me m"Ba

Infinite Gaussian 0.010590 1.6419 2.0 0.6366 1

Truncated Gaussian 0.028727 1.3715 4.0 0.9166 0.8893

Uniform Beam 0.014264 1.1777 1.0 0.9202 1.124

(2) Evaluate nondimensional numbers.

Absorption number NA - Zt

F tSlew number NS

S1SFresnel number NF

tA

Distortion number ND • (2.333 x 10") - w

23

I

(3) Evaluate linear dispersions.

aD2 - 0.5(~

o- 0.5(,

22a 2 3.01(/ C2 x3

Do = -R-

2 o 232r0 a 3.017 (kc2 )-n3/5

2e .8 , 3.7R2,2 or 42 for infinite Gaussian, l/e2

truncated Gaussian, or uniform beam, respectively

D e)

8for 0< 3aT r01 r 0

U- (0) 01 - 1.1890)] for > 3

22oj appropriate to hardware

2a appropriate to hardwareoW

2 2+ 2 2 2L D T j W

(4) Evaluate effective beam qualities.

(M2o2 + a2 + 02)1/2D ( aD oTm M. )

DO

2O ow21/2

I I " (aD/N)

24

'7 -

(5) Evaluate blooming dispersion.

E a+ 0

2Nc -N

B NS + + C2

A = [(Ns + 1) 2 + c 2 1 / 2

[-NDr F~ re ________

-Z 1 (N +lA)(NS +l1OA ] jexp j lj(jn NS- S lA

a 2 a2 HIa

0B "C•C h

(6) Evaluate peak intensity.

P exp(-cZt)lr~t L

I (P) - 2 2 2"RZt(OL + OB) -

~ i

25

LIST OF REFERENCES

1. Breaux, H.J., A Methodology for Developuent of Simple Scaling Laws

for High Energy CW Laser Propagation, Ballistic Research LaboratoryTechnical Report ARBRL-TR-02039, January 1978.

2. Holmes, D.A., J.E. Korba, and P.V. Avizonis, "Pnrametric Study or

Apertured Focused Gaussian Beams", Applied Optics, Vol. 11, p. 565,F

1972.

3. Breaux, H.J., Correlation of Extended Huygens-Fresnel TurbulenceCalculations for a General Class of Tilt Corrected and Uncorrected

Laser Apertures, Ballistic Research Laboratory Interim Memorandum

Report No. 600, May 1978.

4. Braaux, H.J., A Phase Integral Scaling Law Methodology for CombinedRepetitive Pulse and CW High Energy Laser Propagation, Ballistic

Research Laboratory Interim Memorandum Report, August 1977.

5. Wallace, J., and C. Whitney, A Simplified Propagation Code for

Laser Systems Analysis, Charles Stark Draper Laboratory Report

C-4487, June 1977.

6. Klerer, M., and F. Grossman, A New Table for Indefinite Integrals,

Dover, 1971.

7. Zeiders, G.W., A Study of Wave Characteristics Influence on Laser

Selection for Applications: Propacatlon Analysis, W.J. Schaefer

Associates Report WJSA TR-74-18, August 1974.

26